Dilations and Similarity in the Coordinate Plane

Holt McDougal Geometry

Dilations and Similarity in the Coordinate PlaneDilations and Similarity in the Coordinate Plane

Holt Geometry

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz



Dilations and Similarity in the Coordinate Plane

Warm UpSimplify each radical.

1. 2. 3.

Find the distance between each pair of points. Write your answer in simplest radical form.

4. C (1, 6) and D (–2, 0)

5. E(–7, –1) and F(–1, –5)



Apply similarity properties in the coordinate plane.

Use coordinate proof to prove figures similar.

Objectives



dilationscale factor

Vocabulary



A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) (ka, kb).



If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor is less than 1 (k < 1), it is a reduction.

Helpful Hint



Example 1: Computer Graphics Application

Draw the border of the photo after a

dilation with scale factor



Example 1 Continued

RectangleABCD

Step 1 Multiply the vertices of the photo A(0, 0), B(0,

4), C(3, 4), and D(3, 0) by

RectangleA’B’C’D’



Example 1 Continued

Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0).

Draw the rectangle.



Check It Out! Example 1

What if…? Draw the border of the original photo

after a dilation with scale factor



Check It Out! Example 1 Continued Step 1 Multiply the vertices of the photo A(0, 0), B(0,

4), C(3, 4), and D(3, 0) by

RectangleABCD

RectangleA’B’C’D’



Check It Out! Example 1 Continued

Step 2 Plot points A’(0, 0), B’(0, 2), C’(1.5, 2), and D’(1.5, 0).

Draw the rectangle.

A’ D’

B’ C’

01.5

2



Example 2: Finding Coordinates of Similar Triangle

Given that ∆TUO ~ ∆RSO, find the coordinates of U and the scale factor.

Since ∆TUO ~ ∆RSO,

Substitute 12 for RO, 9 for TO, and 16 for OY.

12OU = 144 Cross Products Prop.

OU = 12 Divide both sides by 12.



Example 2 Continued

U lies on the y-axis, so its x-coordinate is 0. Since OU = 12, its y-coordinate must be 12. The coordinates of U are (0, 12).

So the scale factor is




Given that ∆MON ~ ∆POQ and coordinates P (–15, 0), M(–10, 0), and Q(0, –30), find the coordinates of N and the scale factor.

Since ∆MON ~ ∆POQ,

Substitute 10 for OM, 15 for OP, and 30 for OQ.

15 ON = 300 Cross Products Prop.

ON = 20 Divide both sides by 15.




N lies on the y-axis, so its x-coordinate is 0. Since ON = 20, its y-coordinate must be –20. The coordinates of N are (0, –20).

So the scale factor is



Example 3: Proving Triangles Are Similar

Prove: ∆EHJ ~ ∆EFG.

Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2).

Step 1 Plot the points and draw the triangles.



Example 3 Continued

Step 2 Use the Distance Formula to find the side lengths.



Example 3 Continued

Step 3 Find the similarity ratio.

= 2 = 2

Since and E E, by the Reflexive Property,

∆EHJ ~ ∆EFG by SAS ~ .




Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3).

Prove: ∆RST ~ ∆RUV




Step 1 Plot the points and draw the triangles.

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-1

1

2

3

4

5

X

Y

R

ST

UV









Since and R R, by the Reflexive

Property, ∆RST ~ ∆RUV by SAS ~ .



Example 4: Using the SSS Similarity Theorem

Verify that ∆A'B'C' ~ ∆ABC.

Graph the image of ∆ABC

after a dilation with scale

factor



Example 4 Continued

Step 1 Multiply each coordinate by to find the

coordinates of the vertices of ∆A’B’C’.



Step 2 Graph ∆A’B’C’.

Example 4 Continued

C’ (4, 0)

A’ (0, 2)

B’ (2, 4)




Example 4 Continued



Example 4 Continued


Since , ∆ABC ~ ∆A’B’C’ by SSS ~.




Graph the image of ∆MNP after a dilation with scale factor 3. Verify that ∆M'N'P' ~ ∆MNP.




Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.



Step 2 Graph ∆M’N’P’.


-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7-6-5-4-3-2-1

1234567

X

Y









Since , ∆MNP ~ ∆M’N’P’ by SSS ~.



Lesson Quiz: Part I

1. Given X(0, 2), Y(–2, 2), and Z(–2, 0), find the

coordinates of X', Y, and Z' after a dilation with

scale factor –4.

2. ∆JOK ~ ∆LOM. Find the coordinates of M and the scale factor.

X'(0, –8); Y'(8, –8); Z'(8, 0)



Lesson Quiz: Part II

3. Given: A(–1, 0), B(–4, 5), C(2, 2), D(2, –1),

E(–4, 9), and F(8, 3)

Prove: ∆ABC ~ ∆DEF

Therefore, and ∆ABC ~ ∆DEF

by SSS ~.

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Dilations and Similarity in the Coordinate Plane